mixdiff function

Difference of mixture distributions

Density, cumulative distribution function, quantile function and random number generation for the difference of two mixture distributions.

dmixdiff(mix1, mix2, x)

pmixdiff(mix1, mix2, q, lower.tail = TRUE)

qmixdiff(mix1, mix2, p, lower.tail = TRUE)

rmixdiff(mix1, mix2, n)

Arguments

• mix1: first mixture density
• mix2: second mixture density
• x: vector of values for which density values are computed
• q: vector of quantiles for which cumulative probabilities are computed
• lower.tail: logical; if TRUE (default), probabilities are P[X <= x], otherwise P[X > x].
• p: vector of cumulative probabilities for which quantiles are computed
• n: size of random sample

Returns

Respective density, quantile, cumulative density or random numbers.

Details

If $x_1 ~ f_1(x_1)$ and $x_2 ~ f_2(x)$, the density of the difference $d = x_1 - x_2$ is given by

The cumulative distribution function equates to

Both integrals are performed over the full support of the densities and use the numerical integration function integrate.

Examples

# 1. Difference between two beta distributions, i.e. Pr( mix1 - mix2 > 0)
mix1 <- mixbeta(c(1, 11, 4))
mix2 <- mixbeta(c(1, 8, 7))
pmixdiff(mix1, mix2, 0, FALSE)

# Interval probability, i.e. Pr( 0.3 > mix1 - mix2 > 0)
pmixdiff(mix1, mix2, 0.3) - pmixdiff(mix1, mix2, 0)

# 2. two distributions, one of them a mixture
m1 <- mixbeta(c(1, 30, 50))
m2 <- mixbeta(c(0.75, 20, 50), c(0.25, 1, 1))

# random sample of difference
set.seed(23434)
rM <- rmixdiff(m1, m2, 1E4)

# histogram of random numbers and exact density
hist(rM, prob = TRUE, new = TRUE, nclass = 40)
curve(dmixdiff(m1, m2, x), add = TRUE, n = 51)

# threshold probabilities for difference, at 0 and 0.2
pmixdiff(m1, m2, 0)
mean(rM < 0)
pmixdiff(m1, m2, 0.2)
mean(rM < 0.2)

# median of difference
mdn <- qmixdiff(m1, m2, 0.5)
mean(rM < mdn)

# 95%-interval
qmixdiff(m1, m2, c(0.025, 0.975))
quantile(rM, c(0.025, 0.975))